The Distance Formula

In analytic geometry, the distance between two points of the $xy$-plane can be found using the distance formula.  The distance can be from two points on a line or from two points on a line segment.  The distance between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is given by the formula:

$\displaystyle d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

This formula is easily derived by constructing a right triangle with the hypotenuse connecting the two points ($c$) and two legs drawn from the each of the two points to intersect each other ($a$ and $b$), (see image below) and applying the Pythagorean theorem. This theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides . 

Pythagorean Theorem

The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

The image below names the two points, with the distance between them as the variable, $d$.  Notice that the length between each point and the triangle's right angle is found by calculating the difference between the $y$-coordinates and $x$-coordinates, respectively.  The distance formula includes the lengths of the legs of the triangle (normally labeled $a$ and $b$), with the expressions $(y_{2}-y_{1})$ and $(x_{2}-x_{1})$.

Distance Formula

The distance formula between two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, shown as the hypotenuse of a right triangle

Example:  Find the distance between the points $(2,4)$ and $(5,8)$

Substitute the values into the distance formula that is derived from the Pythagorean Theorem:

$\displaystyle d=\sqrt{(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}}$

$\displaystyle \begin{aligned} d&=\sqrt{(5-2)^{2}+(8-4)^{2}}\\&=\sqrt{3^{2}+4^{2}}\\&=\sqrt{25}\\&=5 \end{aligned}$

Midpoint of a Line Segment

In geometry, the midpoint is the middle point of a line segment, or the middle point of two points on a line, and thus is equidistant from both end-points. If you have two points, $(x_{1},y_{1})$ and $(x_{2},y_{2})$, the midpoint of the segment connecting the two points can be found with the formula: 

$\displaystyle (\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2})$

Another way to interpret this formula is an average: we average the $x$-coordinates to find the $x$-coordinate of the midpoint, and we average the $y$-coordinates to find the $y$-coordinate of the midpoint.

By looking at each coordinate, you can see that the $x$-coordinate is halfway between $x_{1}$ and $x_{2}$, and the $y$-coordinate is halfway between $y_{1}$ and $y_{2}$.

Midpoint of a Line Segment

The equation for a midpoint of a line segment with endpoints $(x_{1},y_{1})$and $(x_{2},y_{2})$

Example:  Find the midpoint between $(2,4) $ and $(5,8)$

Substitute the values into the midpoint formula:

$\displaystyle \begin{aligned} (\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2})&=(\frac{2+5}{2}, \frac{4+8}{2})\\&=(\frac{7}{2}, 6) \end{aligned}$